See Appendix REF  for a discussion of our evaluation in comparison to log likelihood. In particular, we emphasize that log likelihood of test samples provides no information about the steady-state distribution of a model [1]}, and that models with high log likelihood exhibit the same misalignment as shortrun and midrun EBMs. We advocate for the use of longrun MCMC samples as a necessary but not sufficient proof of steady-state alignment, and we encourage the field to move away from viewing log likelihood as an indicator of accurate steady-state modeling.
